Sperner proved in 1927 (the paper was published in 1928) his theorem stating that the maximal size of an antichain of subsets of $[n]$ is $\binom{n}{n/2}$. In the introduction to his paper, he mentions that it was Schreier (his advisor) who suggested the question to him. Where did this question come from?

One possibility is Schreier's involvement in proving van der Waerden's theorem on arithmetic sequences. According to van der Waerden, he proved his theorem while visiting Hamburg (where Schreier was based), following a discussion with Schreier and Artin. However, the connection is a bit tenuous.

One of my professors, Gyula Katona, mentions in his doctoral thesis that Sperner used his result to answer the following question: given a square-free integer, what is the maximum number of its positive divisors that do not divide one another? I hope this might help.
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Norbert PintyeFeb 2 '13 at 3:07

Is your question about what motivated Sperner to study antichains?
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john mangualFeb 5 '13 at 19:01

1 Answer
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Deleted while there was a bounty on offer, since it didn't actually answer the question. Undeleted now that the bounty has passed, since the information might be useful to someone else.

Bernhard Beham and Karl Sigmund, A short tale of two cities: Otto Schreier and the
Hamburg-Vienna
connection, The Mathematical Intelligencer, Volume 30, Number 3, 2008, 27-35.

From page 33:

The main content of that seminar was a paper by Witold Hurewicz (1904-1956) and Karl Menger on dimension, which appeared (after many corrections suggested by Schreier) in 1928 in the Mathematische Annalen. By the time it was published, a considerable part of it had been overtaken by events. Indeed, another rising star started to shine in the Artin-Blaschke- Schreier seminar: Emmanuel Sperner (1905-1980) proposed a lemma on the coloring of simplicial decompositions, which greatly simplified the proof of Lebesgue's covering theorem. Sperner's lemma caused Lebesgue's approach to become the most widely used definition of topological dimension (a set is $n$-dimensional if each open cover can be refined so that each point lies in at most $n+1$ open sets). Schreier reported enthusiastically to Menger:

Dear Karl! You will certainly be astonished that I reply so quickly to your kind letter. The main reason is the following: I had recently submitted to our best student, Mr. E. Sperner, the problem to find a nicer proof for Lebesgue's theorem on ${\bf R}^n$. To my pleasure he brought me yesterday an absolutely delicious proof. Since I hope that you will also be happy about it, I will immediately describe to you the proof, which is due to appear in our proceedings . . .

When Sperner submitted his PhD
thesis in 1928, Schreier's report minced no words: 'The following proof has to be qualified as a true work of art.... Finally the invariance of dimension is truly accessible, it follows in a trivial way'.

Gerry, nice as this is, I do not think it addresses the question. The relevant paper here is Ein Satz über Untermengen einer endlichen Menge, while the one discussed in the Math. Intelligencer article is Neuer Beweis für die Invarianz der Dimensionszahl und des Gebiets.
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Andres CaicedoJan 31 '13 at 6:19

According to the papers themselves, Sperner's antichain result dates from early 1927 (February 13, to be exact; published apparently in the first half of 1928), while Sperner's coloring result is from the middle of 1928 (it only says June; the paper itself was published in December).
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Yuval FilmusJan 31 '13 at 6:21

Yes, it's quite possible that I am confusing the two Sperner results. Perhaps the link between them is only in my mind. I'm open to the option of deleting my answer, so it doesn't get awarded a spurious bounty if nothing else shows up.
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Gerry MyersonJan 31 '13 at 6:40

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Gerry, Maybe deleting it while the bounty is active, and undeleting it after?
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Andres CaicedoJan 31 '13 at 7:00